Real numbers

Q: What is the difference between rational and irrational numbers?

A rational number can be written as \dfrac{p}{q} with integers p, q and q \ne 0. An irrational number cannot; its decimal never ends or repeats.

Q: Is the square root of every number irrational?

No. The square root of a perfect square is a whole number (\sqrt{49} = 7). \sqrt{n} is irrational when n is a whole number that is not a perfect square.

Q: What is the Fundamental Theorem of Arithmetic?

Every whole number greater than 1 can be written as a product of primes in exactly one way, apart from the order: 60 = 2^2 \times 3 \times 5.

Q: What does rationalising the denominator mean?

Rewriting a fraction so that the denominator has no square roots, by multiplying top and bottom by a suitable number.

Real numbers are all the numbers on the number line. Rational numbers can be written as a fraction $\frac{p}{q}$ of two integers; their decimals either stop or repeat. Irrational numbers, like $2$ and $π$ , have decimals that never stop and never repeat.

Formulas

A fraction in lowest terms terminates exactly when

q = 2^{m} \times 5^{n}

Euclid's division

a = b q + r, 0 \leq r < b

Two numbers a and b

HCF (a, b) \times LCM (a, b) = a \times b

Worked examples

Example 1. Does

\frac{7}{40}

have a terminating decimal?

$40 = 2^{3} \times 5$ : only 2s and 5s.
So it terminates: $\frac{7}{40} = 0.175$ .

Answer: Yes, 0.175

Example 2. Write

0. \overline{3}

as a fraction.

Let $x = 0.333 \dots$ . Then $10 x = 3.333 \dots$ .
Subtract: $9 x = 3$ , so $x = \frac{3}{9} = \frac{1}{3}$ .

Answer:

\frac{1}{3}

Example 3. Find the HCF of 96 and 404 by Euclid's division.

$404 = 96 \times 4 + 20, 96 = 20 \times 4 + 16$ .
$20 = 16 \times 1 + 4, 16 = 4 \times 4 + 0$ . The last non-zero remainder is 4.

Answer: 4

Example 4. Rationalise the denominator of

\frac{1}{3 + 1}

Multiply top and bottom by $3 - 1$ .
$\frac{3 - 1}{( 3 ) ^{2} - 1 ^{2}} = \frac{3 - 1}{2}$ .

Answer:

\frac{3 - 1}{2}

Common mistakes

Thinking every never-ending decimal is irrational. $0.333 \dots$ repeats, so it is rational $(\frac{1}{3})$ .
Checking the denominator before reducing the fraction. $\frac{21}{30} = \frac{7}{10}$ terminates, even though 30 has a factor 3.
Treating $\frac{22}{7}$ as exactly $π$ . It is only an approximation; $π$ is irrational.

Practice questions with answers

1. Which of the following is a rational number?

A $2$
B $\frac{5}{7}$
Cπ
D $3$

Show answer

Answer: B)

\frac{5}{7}

A rational number can be written as

\frac{p}{q}

with

q

not 0.

\frac{5}{7}

is of this form, so it is rational.

2, 3

and π are irrational.

2. What is the decimal expansion of

\frac{1}{8}

A0.8
B0.18
C0.0125
D0.125

Show answer

Answer: D) 0.125

Divide 1 by 8: 1.000 ÷ 8 = 0.125. The remainder becomes 0, so the expansion terminates.

3. To rationalise the denominator of

\frac{1}{5}

, by what should we multiply both numerator and denominator?

A $5$
B5
C $\frac{1}{5}$
D $10$

Show answer

Answer: A)

5

Multiply numerator and denominator by

5 : \frac{1}{5} = \frac{5}{5}

. The multiplier is

5

4. The number

2

lies on the number line between which two consecutive integers?

A0 and 1
B1 and 2
C2 and 3
D3 and 4

Show answer

Answer: B) 1 and 2

2

is about 1.414. Since 1 < 1.414 < 2, it lies between 1 and 2.

5. Use Euclid's algorithm: HCF(60, 24) = ____.

A6
B12
C24
D120

Show answer

Answer: B) 12

60 =

24 \times 2 + 12

;

24 = 12 \times 2 + 0

. HCF = 12.

6. Which of the following numbers is irrational?

A $2$
B0.75
C $\frac{3}{7}$
D $\frac{16}{4}$

Show answer

Answer: A)

2

2

is non-terminating non-recurring, so it is irrational.

0.75 = \frac{3}{4}, \frac{3}{7}

and

\frac{16}{4} = 4

are all rational.

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Frequently asked questions

What is the difference between rational and irrational numbers?

A rational number can be written as $\frac{p}{q}$ with integers $p, q$ and $q \neq = 0$ . An irrational number cannot; its decimal never ends or repeats.

Is the square root of every number irrational?

No. The square root of a perfect square is a whole number $(49 = 7) . n$ is irrational when $n$ is a whole number that is not a perfect square.

What is the Fundamental Theorem of Arithmetic?

Every whole number greater than 1 can be written as a product of primes in exactly one way, apart from the order: $60 = 2^{2} \times 3 \times 5$ .

What does rationalising the denominator mean?

Rewriting a fraction so that the denominator has no square roots, by multiplying top and bottom by a suitable number.

Real numbers

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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